388(4): Checking the Note by Doug Lindstrom

OK thanks, this is significant progress because Doug’s method gives an additional equation (12). It is a matter of solving Eq. (12) and using omega sub 0 in eq. (13) to give E. I think that the easiest method is to calculate omega sub 0 from the new equation (12) and to use it in Eq. (13) to calculate the total E. This would avoid the problem with an insoluble differential equation. The Lorenz condition is a particular solution of the continuity equation as shown in the previous note. There is a Lorenz gauge as you know. The Lorenz condition is essential to obtain the wave equation from the field equation.

The calculations are o.k., obviously Doug had a sign error in his calculation.
The pricipal question is how fundamental the Lorenz condition (11) is. If we determine phi and A from the wave equations (17,18), then the Lorenz condition is implicitely assumed, insofar it makes sense to apply it to (10), giving the additional constraint (12).
From (13-18) the quantities A, phi and omega are known. So there are two possibilities to determine omega_0:

1) from the vector equation (13) (which gives redundant equations as additional constraints)
2) from the scalar equation (12) which requires that phi be different from zero.

In the second case we have.

Inserting this into (13), we have redundant constraints in vector form again:

This is a complicated diff. eq. for phi (if we want to determine phi from this equation), there is no analytical solution.

Horst

Am 13.09.2017 um 11:39 schrieb EMyrone:

I found Eq. (12), a new and useful antisymmetry equation. There appears to be a sign error in Doug’s derivation and this can be checked by computer algebra. The signs in the second and third terms of my Eq (10) are opposite to Doug’s. Everything else is the same. Doug’s idea is an elegant and important one. The complete set of antisymmetry equations is Eq. (12) to (16). The Lorenz condition (11) is used to find Eq. (12), and to derive the familiar Eqs. (17) and (18). So it has been shown that ECE2 electrodynamics in general rigorously conserves antisymmetry, Q. E. D.